1. Field of the Invention
The present invention relates to a fitting method of model parameters in various types of systems that are described in, for example, differential equations. More concretely, the present invention relates to a program that verifies and controls accuracy of model parameter computation using quantifier elimination method.
2. Description of the Related Art
In the analysis of various types of systems described in, for example, ordinary differential equations, a parameter fitting method of fitting the model parameters and the initial values of differential equations is executed in such a way that the observed values of a time-series become equal to the commutated values of the time-series of a differential equation model. For example, in order to clarify the mechanism of a biochemical reaction, a fitting computation is executed in such a way that the observed values of a time-series and the values computed by a differential equation model are the same, thereby estimating model parameters and initial values.
According to this method, in the first step, the time-series computations are executed at first by a differential equation model using suitable model parameters and suitable initial values. In the second step, model parameters and initial values are estimated in such a way that the results of computation and the observed values of a time-series match with each other. In the third step, the time-series computations are executed by a differential equation model using the estimated model parameters and the estimated initial values. Then, the second and third steps are repeated until the sum of squared residuals among the results of the time-series computations and the observed values of a time-series becomes minimum, or equal to a certain threshold value or less.
As a conventional technology of such a parameter fitting method, there is the following document related to the mechanism analysis system of HIV proteinase.
[Nonpatent Literature 1]
Hermann Georg Holzhutter and Alfredo Colosimo; SIMFIT: a microcomputer software-toolkit for modelistic studies in biochemistry, CABIOS Vol. 6, No. 1, pp. 23-28 (1990) (http://www.gepasi.org/gep3tuts.html.)
In this literature, the following steps are executed.    (1) Time-series simulations are executed by a simulator using suitable initial values and suitable initial parameters.    (2) In order to match the results of the simulation with the experimental values, a weighted sum of squared residuals is computed using observed values, the results of simulation and model parameter values by a minimizer and then model parameters are computed in such a way that the computed sum becomes minimum.    (3) Time-series simulations are executed by the simulator using the computed model parameters. At this time, initial values are changed so that the initial values such that the weighted sum of squared residuals computed by the minimizer becomes minimum are detected.    (4) Time-series simulations are executed by the simulator using the model parameters computed in (2) and the initial values detected in (3).    (5) A weighted sum of squared residuals is computed by the minimizer using the observed values, the simulation results and the model parameters and then model parameters for minimizing the value are obtained.    (6) Returning to (2) until the model parameters and the initial values converge up to the predetermined range. In this way, the accuracies of the values of model parameters that are obtained by a fitting computation is discussed using statistic data such as a sum of squared residuals in the conventional technology. Therefore, in the conventional fitting computation, the influence caused by the difference between the accuracy of the time-series computations and that of a fitting computation or the fluctuation among observations is present. Accordingly, there arises a problem such that the accuracy of model parameters or that of initial values that are obtained by the fitting computation cannot be correctly discussed. That is, according to the conventional method, there are the following problems. Firstly, it cannot be distinguished whether the standard deviation of model parameters that is obtained by fitting the observed values of a time-series and the differential equation model is generated by an experimental error or a numerical calculation. Therefore, the accuracy of the obtained values of parameters cannot be discussed. Furthermore, there is no method of easily clarifying the factors of computation accuracy or controlling the accuracy.